scholarly journals Higher-Order Accurate Finite Volume Discretization of the Three-Dimensional Poisson Equation Based on An Equation Error Method

2018 ◽  
Vol 6 (6) ◽  
pp. 107-123
Author(s):  
Yaw Kyei
Keyword(s):  
Finite Volume ◽  
Three Dimensional ◽  
Integral Form ◽  
Higher Order ◽  
Finite Volume Schemes ◽  
Finite Volume Discretization ◽  
Equation Error ◽  
Error Expansion ◽  
Control Volumes ◽  
Physical Phenomena

Efficient higher-order accurate finite volume schemes are developed for the threedimensional Poisson’s equation based on optimizations of an equation error expansion on local control volumes. A weighted quadrature of local compact fluxes and the flux integral form of the equation are utilized to formulate the local equation error expansions. Efficient quadrature weights for the schemes are then determined through a minimization of the error expansion for higher-order accurate discretizations of the equation. Consequently, the leading numerical viscosity coefficients are more accurately and completely determined to optimize the weight parameters for uniform higher-order convergence suitable for effective numerical modeling of physical phenomena. Effectiveness of the schemes are evaluated through the solution of the associated eigenvalue problem. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.

scholarly journals Extending the global-direction stencil with "face-area-weighted centroid" to unstructured finite volume discretization from integral form

2020 ◽  
Author(s):  
Lingfa Kong ◽  
Yidao Dong ◽  
Wei Liu ◽  
Huaibao Zhang
Keyword(s):  
Finite Volume ◽  
Differential Form ◽  
Integral Form ◽  
Finite Volume Methods ◽  
Higher Order ◽  
Reconstruction Method ◽  
Computational Accuracy ◽  
Face Area ◽  
Finite Volume Discretization ◽  
Numerical Performance

Abstract Accuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction. For the commonly used k-exact reconstruction method, the cell centroid is always chosen as the reference point to formulate the reconstructed function. But in some practical problems, such as the boundary layer, cells in this area are always set with high aspect ratio to improve the local field resolution, and if geometric centroid is still utilized for the spatial discretization, the severe grid skewness cannot be avoided, which is adverse to the numerical performance of unstructured finite volume solver. In previous work, we explored a novel global-direction stencil and combine it with face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy. Note, however, that the differential form is hard to achieve higher-order accuracy, and in order to set stage for the method promotion on higher-order numerical simulation, in this research, we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization in integral form. Numerical examples governed by linear convective, Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension. Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid, the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid. As a result, on unstructured finite volume discretization from integral form, the method also has a better numerical performance.

scholarly journals Extending the global-direction stencil with “face-area-weighted centroid” to unstructured finite volume discretization from integral form

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Lingfa Kong ◽  
Yidao Dong ◽  
Wei Liu ◽  
Huaibao Zhang
Keyword(s):  
Convergence Rate ◽  
Finite Volume ◽  
Differential Form ◽  
Integral Form ◽  
Finite Volume Methods ◽  
Reconstruction Method ◽  
Computational Accuracy ◽  
Face Area ◽  
Finite Volume Discretization ◽  
Skewness Reduction

AbstractAccuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction. For the commonly used k-exact reconstruction method, the cell centroid is always chosen as the reference point to formulate the reconstructed function. But in some practical problems, such as the boundary layer, cells in this area are always set with high aspect ratio to improve the local field resolution, and if geometric centroid is still utilized for the spatial discretization, the severe grid skewness cannot be avoided, which is adverse to the numerical performance of unstructured finite volume solver. In previous work [Kong, et al. Chin Phys B 29(10):100203, 2020], we explored a novel global-direction stencil and combined it with the face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy. Greatly inspired by the differential form, in this research, we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization from integral form on both second and third-order finite volume solver. Numerical examples governed by linear convective, Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension. Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid, the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid. As a result, on unstructured finite volume discretization from integral form, the method also has superiorities on both computational accuracy and convergence rate.

scholarly journals A Higher-Order Chimera Method for Finite Volume Schemes

2017 ◽  
Vol 25 (3) ◽  
pp. 691-706 ◽  
Author(s):  
Luis Ramírez ◽  
Xesús Nogueira ◽  
Pablo Ouro ◽  
Fermín Navarrina ◽  
Sofiane Khelladi ◽  
...  
Keyword(s):  
Finite Volume ◽  
Higher Order ◽  
Finite Volume Schemes

CONVERGENCE OF A FINITE VOLUME SCHEME FOR GAS–WATER FLOW IN A MULTI-DIMENSIONAL POROUS MEDIUM

2013 ◽  
Vol 24 (01) ◽  
pp. 145-185 ◽  
Author(s):  
MOSTAFA BENDAHMANE ◽  
ZIAD KHALIL ◽  
MAZEN SAAD
Keyword(s):  
Porous Medium ◽  
Finite Volume ◽  
Three Dimensional ◽  
Energy Estimates ◽  
Finite Volume Scheme ◽  
Convergence Properties ◽  
Finite Volume Schemes ◽  
Finite Element Scheme ◽  
Discrete Energy ◽  
Volume Scheme

This paper deals with construction and convergence analysis of a finite volume scheme for compressible/incompressible (gas–water) flows in porous media. The convergence properties of finite volume schemes or finite element scheme are only known for incompressible fluids. We present a new result of convergence in a two or three dimensional porous medium and under the only consideration that the density of gas depends on global pressure. In comparison with incompressible fluid, compressible fluids requires more powerful techniques; especially the discrete energy estimates are not standard.

Error estimates for higher-order finite volume schemes for convection–diffusion problems

2018 ◽  
Vol 26 (1) ◽  
pp. 35-62
Author(s):  
Dietmar Kröner ◽  
Mirko Rokyta
Keyword(s):  
Error Estimates ◽  
Finite Volume ◽  
Original Problem ◽  
A Priori ◽  
Higher Order ◽  
Finite Volume Schemes ◽  
Convection Diffusion ◽  
Type Reconstruction ◽  
Convection Dominated ◽  
Diffusion Problems

AbstractIt is still an open problem to provea priorierror estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domainΩin ℝ2and we can prove such kind of ana priorierror estimate. In the part of the estimate, which refers to the discretization of the convective term, we gainh1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.

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